Let `S_(1) : x^(2) + y^(2) = 25` and `S_(2) : x^(2) + y^(2) - 2x -2y - 14 = 0` be two circles.
STATEMENT-1 : `S_(1) and S_(2)` have exactly two common tangents.
and
STATEMENT-2 : If two circles touches each other internally they have one common tangent.

Options

A. Statement-1 is true, statement-2 is true, Statement -2 is a correct explanation for Statement -1
B. Statement -1 is true, Statement-2 is true , Statement-2 is NOT a correct explanation for statement-1
C. Statement-1 is true, Statement-2 is False
D. Statement-1 is False, Statement-2 is true

Answer

To solve the problem, we need to analyze the two circles given by the equations \( S_1: x^2 + y^2 = 25 \) and \( S_2: x^2 + y^2 - 2x - 2y - 14 = 0 \).

### Step 1: Identify the first circle \( S_1 \)
The equation of the first circle is:
\[ S_1: x^2 + y^2 = 25 \]
From this equation, we can identify:
- Center \( C_1 = (0, 0) \)
- Radius \( r_1 = \sqrt{25} = 5 \)

### Step 2: Rewrite the second circle \( S_2 \)
The equation of the second circle is:
\[ S_2: x^2 + y^2 - 2x - 2y - 14 = 0 \]
We can rearrange this equation to find the center and radius. First, we complete the square for \( x \) and \( y \).

1. Rearranging the equation:
\[ x^2 - 2x + y^2 - 2y = 14 \]
2. Completing the square:
- For \( x^2 - 2x \):
\[ x^2 - 2x = (x - 1)^2 - 1 \]
- For \( y^2 - 2y \):
\[ y^2 - 2y = (y - 1)^2 - 1 \]
3. Substituting back:
\[ (x - 1)^2 - 1 + (y - 1)^2 - 1 = 14 \]
\[ (x - 1)^2 + (y - 1)^2 = 16 \]

From this, we can identify:
- Center \( C_2 = (1, 1) \)
- Radius \( r_2 = \sqrt{16} = 4 \)

### Step 3: Analyze the positions of the circles
Now we have:
- Circle \( S_1 \) with center \( (0, 0) \) and radius \( 5 \)
- Circle \( S_2 \) with center \( (1, 1) \) and radius \( 4 \)

Next, we calculate the distance between the centers \( C_1 \) and \( C_2 \):
\[ d = \sqrt{(1 - 0)^2 + (1 - 0)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 4: Determine the relationship between the circles
To determine the number of common tangents, we compare the distance \( d \) with the sum and difference of the radii:
- Sum of the radii: \( r_1 + r_2 = 5 + 4 = 9 \)
- Difference of the radii: \( |r_1 - r_2| = |5 - 4| = 1 \)

Since \( d = \sqrt{2} \) is less than \( r_1 + r_2 \) (9) and greater than \( |r_1 - r_2| \) (1), the circles intersect at two points. Therefore, they have exactly two common tangents.

### Conclusion for Statement 1
**Statement 1**: \( S_1 \) and \( S_2 \) have exactly two common tangents. This statement is **true**.

### Step 5: Analyze Statement 2
**Statement 2**: If two circles touch each other internally, they have one common tangent.

For two circles to touch internally, the distance between their centers \( d \) must equal the difference of their radii \( |r_1 - r_2| \). In this case, there is exactly one common tangent.

### Conclusion for Statement 2
**Statement 2** is also **true**.

### Final Answer
Both statements are true, but they describe different scenarios. Thus, we conclude:
- Statement 1 is true.
- Statement 2 is true.